We assume that the mucus layer in this geometry has a negligible thickness relatively to the airways radii [1]}, [2]}.
Then, the principal curvatures of the air–Bingham fluid interface in the generation \(i\)  can be assimilated to the cylindrical airway principal curvatures: \(1/r_i\)  in the radial direction and 0 in the axial direction.
These curvatures induce a Laplace pressure drop \(p_{L,i}\)  between the air and the Bingham fluid that formulates into
\(p_{L,i} = - \frac{\gamma }{r_i}\) 
